This site is being phased out.

Approximating paths

From Mathematics Is A Science
Jump to navigationJump to search

Theorem. In an open region $R \subset {\bf R}^n$, if two points are connected by a path, they can be connected by a step-path with edges parallel to the coordinate axes (a "discrete curve").

StepApproximation.png

We start with, what if $R$ is a disk?

Since it's round, the solution is simple:

Homework2Explanation1.png

Lemma. In a disk, we can always get from $(a,b)$ to $(c,d)$ by a step-path with no more than $4$ steps: $$(a,b)\rightarrow (a,0) \rightarrow (0,0)\rightarrow (c,0)\rightarrow (c,d).$$

Based on the lemma, the construction for a general $R$ looks like this.

We cover the path between $A$ and $B$ with disks $D_1,...,D_n$, within $R$, and then apply the lemma $n$ times:

  • from $A=A_0$ to any $A_1 \in D_1 \cap D_2$ within disk $D_1$,
  • then from $A_1$ to any $A_2 \in D_2 \cap D_3$ within disk $D_2$,
  • $\ldots$,
  • from $A_{n-1}$ to $A_n=B$ within disk $D_n$.

This should work but how do we know that we can get there in a finite number of steps?

This is not a problem, because...

Homework2Explanation2.png

For each point on the path, find an open disk in $R$ centered at that point. The path, image of $[0,1]$ under a continuous $p$, is compact, therefore this cover has a finite subcover... etc. (Exercise)

Exercise. Find a proof that doesn't invoke compactness or other point-set topology concepts.